Recent zbMATH articles in MSC 46Fhttps://www.zbmath.org/atom/cc/46F2021-04-16T16:22:00+00:00WerkzeugProperties of field functionals and characterization of local functionals.https://www.zbmath.org/1456.812962021-04-16T16:22:00+00:00"Brouder, Christian"https://www.zbmath.org/authors/?q=ai:brouder.christian"Dang, Nguyen Viet"https://www.zbmath.org/authors/?q=ai:dang.nguyen-viet"Laurent-Gengoux, Camille"https://www.zbmath.org/authors/?q=ai:laurent-gengoux.camille"Rejzner, Kasia"https://www.zbmath.org/authors/?q=ai:rejzner.kasiaSummary: Functionals (i.e., functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the proper space of test functions (smooth functions) and of the relevant concept of differential (Bastiani differential) are discussed. The relation between the multiple derivatives of a functional and the corresponding distributions is described in detail. It is proved that, in a neighborhood of every test function, the support of a smooth functional is uniformly compactly supported and the order of the corresponding distribution is uniformly bounded. Relying on a recent work by Dabrowski, several spaces of functionals are furnished with a complete and nuclear topology. In view of physical applications, it is shown that most formal manipulations can be given a rigorous meaning. A new concept of local functionals is proposed and two characterizations of them are given: the first one uses the additivity (or Hammerstein) property, the second one is a variant of Peetre's theorem. Finally, the first step of a cohomological approach to quantum field theory is carried out by proving a global Poincaré lemma and defining multi-vector fields and graded functionals within our framework.{
\copyright 2018 American Institute of Physics}Factorization in Denjoy-Carleman classes associated to representations of \((\mathbb{R}^d, +)\).https://www.zbmath.org/1456.460392021-04-16T16:22:00+00:00"Debrouwere, Andreas"https://www.zbmath.org/authors/?q=ai:debrouwere.andreas"Prangoski, Bojan"https://www.zbmath.org/authors/?q=ai:prangoski.bojan"Vindas, Jasson"https://www.zbmath.org/authors/?q=ai:vindas.jassonThe authors introduce Denjoy-Carleman classes of ultradifferentiable vectors for two types of moderate growth representations of \((\mathbb{R}^d,+)\) on sequentially complete locally convex Hausdorff spaces and they prove a~strong factorization theorem of Dixmier-Malliavin type for these classes.
This factorization theorem solves Conjecture 6.4 of \textit{H. Gimperlein} et al. [J. Funct. Anal. 262, No. 2, 667--681 (2012; Zbl 1234.22006)]. The main difference between the present approach and the previous works in this direction, which allows the authors to prove the strong instead of the weak factorization property, is to consider general Fourier multipliers rather than infinite order differential operators.
As an application of their results, they show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property. The problem of factorization of convolution algebras of smooth functions emerged from \textit{L. Ehrenpreis}' work in 1960 [Am. J. Math. 82, 522--588 (1960; Zbl 0098.08401)] on fundamental solutions of convolution operators. He asked whether \(\mathcal{D}(\mathbb{R}^d)\) has the strong factorization property. Rubel, Squires and B. A. Taylor in 1978 [\textit{L. A. Rubel} et al., Ann. Math. (2) 108, 553--567 (1978; Zbl 0402.32002)] showed that \(\mathcal{D}(\mathbb{R}^3)\) does not have the strong factorization property and they proved that \(\mathcal{D}(\mathbb{R}^d)\) always satisfies the weak factorization property. \textit{J. Dixmier} and \textit{P. Malliavin} [Bull. Sci. Math., II. Sér. 102, 305--330 (1978; Zbl 0392.43013)] gave a negative answer to Ehrenpreis' question for \(d = 2\). Finally, the problem was completely settled by \textit{R. S. Yulmukhametov} [Sb. Math. 190, No. 4, 597--629 (1999; Zbl 0940.30030); translation from Mat. Sb. 190, No. 4, 123--157 (1999)] showing that the space \(\mathcal{D}(\mathbb{R})\) satisfies the strong factorization property. \textit{K.-i. Miyazaki} [J. Sci. Hiroshima Univ., Ser. A 24, 527--533 (1960; Zbl 0099.32302)], \textit{H. Petzeltova} and \textit{P. Vrbova} [Commentat. Math. Univ. Carol. 19, 489--499 (1978; Zbl 0382.46014)], and \textit{J. Voigt} [Stud. Math. 77, 333--348 (1984; Zbl 0491.46034)] independently proved that the Schwartz space of rapidly decreasing smooth functions has the strong factorization property.
Reviewer: José Bonet (Valencia)Lipschitz estimates for functions of Dirac and Schrödinger operators.https://www.zbmath.org/1456.811742021-04-16T16:22:00+00:00"Skripka, A."https://www.zbmath.org/authors/?q=ai:skripka.a-n|skripka.annaSummary: We establish new Lipschitz-type bounds for functions of operators with noncompact perturbations that produce Schatten class resolvent differences. The results apply to suitable perturbations of Dirac and Schrödinger operators, including some long-range and random potentials, and to important classes of test functions. The key feature of these bounds is an explicit dependence on the Lipschitz seminorm and decay parameters of the respective scalar functions and, in the case of Dirac and Schrödinger operators, on the \(L^p\)- or \(\mathcal{l}^p(L^2)\)-norm of the potential.
{\copyright 2021 American Institute of Physics}A course in analysis. Vol. V: Functional analysis, some operator theory, theory of distributions.https://www.zbmath.org/1456.460012021-04-16T16:22:00+00:00"Jacob, Niels"https://www.zbmath.org/authors/?q=ai:jacob.niels"Evans, Kristian P."https://www.zbmath.org/authors/?q=ai:evans.kristian-pThis fifth instalment of the multi-volume Course in Analysis (the previous parts are [Zbl 1327.26012], [Zbl 1401.26001], [Zbl 1381.28002] and [Zbl 1407.00004]) is dedicated to three main themes: functional analysis, (a non-comprehensive presentation of) operator theory, and distributions, continuing in an organic manner its preceding parts. The high standards of this series are successfully maintained, their main attributes (clarity of the presentation, perfectly adapted depth of the material for a wide readership, excellently chosen problems for illustrating the presented theory, etc.) being easily recognizable in the current volume as well. The rich experience of the authors in teaching analysis (in the broadest sense) transpires on every page and I highly recommend this book to anyone teaching or studying the topics presented.
The more than 850-pages long volume is divided into three parts dedicated to the main themes mentioned above, containing also a preface, an introduction, a list of symbols as well as three appendices, a symbol index, a continuation of the list of contributors to analysis from the first four volumes as well as solutions to the proposed problems and the standard comprehensive list of references.
In the first (and main, given its topic and length) part of the book the reader takes a 14-stop journey (mimicking maybe the 14~chapters of the part dedicated to Fourier Analysis in the previous volume of this series) in the world of Functional Analysis, dealing with topics such as infinite-dimensional vector spaces (in particular, topological vector spaces, Banach and Hilbert spaces as well as dual spaces), linear functionals, linear, adjoint and unbounded operators, Fredholm theory, spectral theory (including the Gelfand-Naimark theory) and self-adjoint operators, convexity and integral representations as well as various themes grouped under the headline ``Selected topics''.
The second part contains altogether seven~chapters on, as the authors put it, ``some'' operator theory, where integral operators, one-parameter semigroups of operators, positivity preserving operators and Markovian semigroups, regular Sturm-Liouville problems, a brief introduction to Sobolev spaces, operators induced by the Dirichlet problem, and, again, some selected topics are presented.
This is followed by a six-chapter third part on distribution theory, dealing with subjects such as function spaces (in particular Fréchet spaces), distributions in the sense of Schwartz, tempered distributions and the Fourier transform, tensor products, kernels, and Calderón-Zygmund operators. The three appendices are dedicated to completeness, nets and the Riesz representation theorem.
Like the previous volumes of the series, the present one is logically constructed and, by paying great attention to details (such as suitable examples for justifying generalizations, remarks stressing surprising connections and properties, suitable problems for illustrating the theory, etc.), provides a valuable resource for teaching material, too. The book is suitable for advanced undergraduate students in mathematics with a keen interest in (functional) analysis, for graduate students in analysis, ordinary differential equations or operator theory, and also as a reference for researchers. I am looking forward to reading the next books in this excellent series which creates a world, if such an unorthodox comparison be allowed, somehow similar in complexity to the (likewise lengthy) ``Game of Thrones'' saga. As previously announced, they shall be dealing with theory of partial differential equations, differential geometry, differentiable manifolds, and Lie groups.
Reviewer: Sorin-Mihai Grad (Wien)Regular generalized solutions to semilinear wave equations.https://www.zbmath.org/1456.350782021-04-16T16:22:00+00:00"Deguchi, Hideo"https://www.zbmath.org/authors/?q=ai:deguchi.hideo"Oberguggenberger, Michael"https://www.zbmath.org/authors/?q=ai:oberguggenberger.michael-bSummary: The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error.Generalized Schwartz type spaces and LCT based pseudodifferential operator.https://www.zbmath.org/1456.352492021-04-16T16:22:00+00:00"Jain, Pankaj"https://www.zbmath.org/authors/?q=ai:jain.pankaj"Kumar, Rajender"https://www.zbmath.org/authors/?q=ai:kumar.rajender"Prasad, Akhilesh"https://www.zbmath.org/authors/?q=ai:prasad.akhileshSummary: In connection with the LCT, in this paper, we define the Schwartz type spaces \(\mathcal{S}_{\Delta,\alpha,A} \), \(\mathcal{S}^{\Delta,\beta,B}\), \(\mathcal{S}^{\Delta,\beta,B}_{\Delta,\alpha,A} \), and study the mapping properties of LCT between these spaces. Moreover, we define a generalized \(\Delta\)-pseudo differential operator and investigate its mapping properties in the framework of the above Schwartz type spaces.Radial and angular derivatives of distributions.https://www.zbmath.org/1456.460382021-04-16T16:22:00+00:00"Brackx, Fred"https://www.zbmath.org/authors/?q=ai:brackx.fred-fSummary: When expressing a distribution in Euclidean space in spherical co-ordinates, derivation with respect to the radial and angular co-ordinates is far from trivial. Exploring the possibilities of defining a radial derivative of the delta distribution \(\delta ( \underline{x})\) (the angular derivatives of \(\delta ( \underline{x})\) being zero since the delta distribution is itself radial) led to the introduction of a new kind of distributions, the so-called signumdistributions, as continuous linear functionals on a space of test functions showing a singularity at the origin. In this paper we search for a definition of the radial and angular derivatives of a general standard distribution and again, as expected, we are inevitably led to consider signumdistributions. Although these signumdistributions provide an adequate framework for the actions on distributions aimed at, it turns out that the derivation with respect to the radial distance of a general (signum) distribution is still not yet unambiguous.
For the entire collection see [Zbl 1432.30001].Laplace transform of functions defined on a bounded interval.https://www.zbmath.org/1456.440022021-04-16T16:22:00+00:00"Stanković, Bogoljub"https://www.zbmath.org/authors/?q=ai:stankovic.bogoljubSummary: Laplace transform \(\dot{\mathcal L}\) for functions belonging to \(L[0,b]\), \(0< b < \infty\) is defined. This definition is given by using the idea of \textit{H. Komatsu} [J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 805--820 (1987; Zbl 0644.44001)] and [in: Structure of solutions of differential equations. Proceedings of the Taniguchi symposium, Katata, Japan, June 26--30, 1995 and the RIMS symposium, Kyoto, Japan, July 3--7, 1995. Singapore: World Scientific. 227--252 (1996; Zbl 0894.35005)] for Laplace hyperfunctions. As an application of \(\dot{\mathcal L}\) we solve an equation with fractional derivative and an integral equation of the first kind of convolution type.